When I asked a mohel, that made bris to my son, what he is doing with orlot after the bris, he told me that he is burying them at his garden. Then he smiled and told that the plants grow very well in his garden :)
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jutkyOct 2 '11 at 20:31

michael, welcome to Judaism.SE, and thanks very much for these source! I merged your second answer into your first; in the future, you can do this yourself by editing. Also, please consider registering your account, which will give you access to more of the site's features.
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Isaac MosesApr 29 '12 at 14:17

A family member was in an accident and there were unidentifiable pieces of flesh and skin preserved in formaldehyde which a rav told us must be buried. so the chevra kadisha did it for us.
i dont know about internal organs but I've never heard of burying that.

The majority opinion follows Rabbi Yechezkel ben Yehuda Landau (1713 – 1793) opinion at Noda B’Yehuda I, Yoreh Deah (YD) 90, who holds that the mitzvah to bury separated body parts is required only of people who are dead, because it would be a disgrace not to. Rav Moshe Feinstein (1895-1986), however, held that even the body parts of living people must be buried because a kohen could become tamei from contact with those parts. Igros Moshe, YD I, 231, citing Nazir 43b. See also, Rambam (Rabbi Moshe ben Maimon, aka Maimonedes (1135-1204)), Tumas Mes 2:3. In Berachos 5b and Baba Basra 116a, Rabbi Yochanan is quoted as saying that he was carrying with him a "bone" of his deceased 10th son. Rashi believes the bone was small enough that it would not convey tumah. The Rashbam (at Baba Basra 116a) argues that Rabbi Yochanan must have been carrying his son's tooth, since that definitely would not convey tumah and does not require burial. But that position is in dispute. Although it is well held that a person may benefit from the tooth of a living person, others question whether one is permitted benefit from the tooth of a deceased person. See, e.g., Ran (Rav Nissim of Gerona, 1320-1376), Chulin 122a; Rabbi Ovadia Yosef (b. 1920), Shlita, Yabia Omer, III, YD 21.

I heard a lecture from Rabbi Elchanan Zohn, head of the Chevra Kadisha of Queens, about this.

Something like a small tumor, or tonsils, would certainly not need burial.

A limb -- flesh, bone, and sinew -- such as a finger would require burial.

For internal organs, the opinion heard orally from Rabbi Moshe Feinstein -- and repeatedly from his son Rabbi Dovid Feinstein shlit'a -- is that any organ that holds something requires burial -- even a gallbladder. The prevalent practice of many, many poskim, however, is that they don't.

For the former -- it says Yiftach was buried "in the cities of Gilad", medrish says his limbs fell off one by one and were buried in various cities.
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ShalomSep 11 '11 at 10:10

For internal organs -- as I said it's debated by contemporary poskim, I'm quoting an instance I was aware of; no I don't know the exact reasoning behind their sources beyond that.
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ShalomSep 11 '11 at 10:11

I don't know if this is the same thing, but here is a letter the Lubavitcher Rebbe wrote in 1965 to the Association of Orthodox Doctors. In the letter, the Rebbe encourages religious doctors to incorporate Halacha into their medical practices. One of the issues the Rebbe discusses is autopsies. Here is what he writes:

For determining the cause of death: In most instances, this knowledge is not vital at all.

In a situation where it is necessary to save another life on the spot (such as exonerating someone who is accused of poisoning by doing an autopsy, etc.) — they should make incisions only where absolutely required, and bury all of the body parts afterwards.

This is not quite the same: here we are talking about parts of the body after death, while my question was focusing on those removed while the person is still alive.
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yydlOct 2 '11 at 18:03

@yyd: True, but limbs that have been removed/fallen off from a live body can cause impurity just like the limb of a dead body, so there are some parallels. See last Rashi on Sotah 3A: e-daf.com/index.asp?ID=2358&size=1 (don't have a halachic source right now)
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MenachemOct 2 '11 at 18:16

It says by Yiftach that he was buried in the cities of Gilad. The gemarah explains that he contracted a disease which made him lose limbs over time which were then buried in different cities in which he lost them.